Fluid Mechanics

ME 2592013-2014

Introduction

• Course Aim– Providing necessary knowledge of fluid properties

and forces on different surfaces – Introduce different types of flows and

implementation using the conservation equations– Introduce diemensional analysis

Introduction

• Teaching and learning methods – Lectures– Tutorial – Lab

• Student assessments– 9th week exam (written) 20%– Continues assessment (in class & homework,

quizzes) 30%– Final assessment (written) 50%

Introduction

• Course content – Introduction to fluid properties – Fluid statics– Fluid dynamics– Dynamics of an incompressible flow– Flow of incompressible fluids through pipes– Dimensional analysis and PI theorem

Introduction

• Required books– Frank M. White, “Fluid Mechanics”, McGraw-Hill,

international edition, sixth Edition,2010

Chapter one

Introduction to fluid mechanics

SEQUENCE OF CHAPTER 1•Introduction•Objectives•1.1 Fluid Concept•1.2 Units and Dimensions•1.3 Fluid Continuum•1.4 Flow Patterns•1.5 Fluid Density•1.6 Viscosity•1.7 Surface Tension•1.8 Vapour Pressure•Summary

Introduction This chapter will begin with several concepts,

definition, terminologies and approaches which should be understood by the students before continuing reading the rest of this module.

Then, it introduces the student with typical properties of fluid and their dimensions which are then being used extensively in the next chapters and units like pressure, velocity, density and viscosity.

Some of these can be used to classify type and characteristic of fluid, such as whether a fluid is incompressible or not or whether the fluid is Newtonian or non-Newtonian.

Objectives• At the end of this chapter, you should be

able to :

explain the concept of fluid continuity and flow representations using streamlines, streaklines and pathlines,

Identify and describe typical fluid properties and their units and dimensions

1.1 Fluid Concept• Fluid mechanics is a division in applied mechanics

related to the behaviour of liquid or gas which is either in rest or in motion.

• The study related to a fluid in rest or stationary is referred to fluid static, otherwise it is referred to as fluid dynamic.

• Fluid can be defined as a substance which can deform continuously when being subjected to shear stress at any magnitude. In other words, it can flow continuously as a result of shearing action. This includes any liquid or gas.

1.1 Fluid Concept• Thus, with exception to solids, any other matters can

be categorised as fluid. In microscopic point of view, this concept corresponds to loose or very loose bonding between molecules of liquid or gas, respectively.

• Examples of typical fluid used in engineering applications are water, oil and air.

• An analogy of how to understand different bonding in solids and fluids is depicted in Fig. 1.1

1.1 Fluid Concept

Figure 1.1 Comparison Between Solids, Liquids and Gases

• For solid, imagine that the molecules can be fictitiously linked to each other with springs.

(a) Solid (b) Liquid (c) Gas

k

k k

k

Free surface

1.1 Fluid Concept In fluid, the molecules can move freely but are

constrained through a traction force called cohesion. This force is interchangeable from one molecule to another.

For gases, it is very weak which enables the gas to disintegrate and move away from its container.

For liquids, it is stronger which is sufficient enough to hold the molecule together and can withstand high compression, which is suitable for application as hydraulic fluid such as oil. On the surface, the cohesion forms a resultant force directed into the liquid region and the combination of cohesion forces between adjacent molecules from a tensioned membrane known as free surface.

1.2 Units and Dimensions• The primary quantities which are also referred to as

basic dimensions, such as L for length, T for time, M for mass and Q for temperature.

• This dimension system is known as the MLT system where it can be used to provide qualitative description for secondary quantities, or derived dimensions, such as area (L2), velocity (LT-1) and density (ML-3).

• In some countries, the FLT system is also used, where the quantity F stands for force.

1.2 Units and Dimensions• An example is a kinematic equation for the velocity V

of a uniformly accelerated body,

• V = V0 + at

where V0 is the initial velocity, a the acceleration and t the time interval. In terms for dimensions of the equation, we can expand that

LT-1 = LT -1 + LT-2 • T

Example 1.1 The free vibration of a particle can be simulated by the

following differential equation:

where m is mass, u is velocity, t is time and x is displacement. Determine the dimension for the stiffness variable k.

0 kxdt

dum

Example 1.1 By making the dimension of the first term equal to the

second term:

[m] • = [k]•[x]

Hence,

[k] = =

= MT-2

[ u ]

[ t ]

[ m ] • [ u ]

[ t ] • [ x ]

M • LT-1

LT

1.3 Fluid Continuum• Since the fluid flows continuously, any method and

technique developed to analyse flow problems should take into consideration the continuity of the fluid. There are two types of approaches that can be used:

1.Eulerian approach — analysis is performed by defining a control volume to represent fluid domain which allows the fluid to flow across the volume. This approach is more appropriate to be used in fluid mechanics.

2.Lagrangian approach — analysis is performed by tracking down all motion parameters and deformation of a domain as it moves. This approach is more suitable and widely used for particle and solid mechanics.

1.3 Fluid Continuum The fluid behaviour in which its properties are

continuous field variables, either scalar or vector, throughout the control volume is known as continuum. From this concept, several fluid or flow definitions can be made as follows:

Steady state flow — A flow is said to be in steady state if its properties is only a function of position (x,y,z) but not time t:

x,y,z), V = V x,y,z)

An example is the velocity of a steady flow of a river where the upstream and downstream velocities are different but their values does not change through time.

1.3 Fluid Continuum• Uniform flow — A flow is said to be uniform if its

velocity and all velocity components is only a function of time t:

• V = V t)An example is the air flow in a constant diameter duct where the velocity is constant throughout the length of the duct but can be increased uniformly by increasing the power of the fan.

• Isotropic fluid — A fluid is said to be isotropic if its density is not a function of position (x,y,z) but may vary with time t:

t) An example is the density of a gas in a closed container where the container is heated. The density is constant inside the container but gradually increases with time as the temperature increases.

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1.4 Flow Patterns• The three ways to represent fluid flow:

1. Streamlines — A streamline is formed by tangents of the velocity field of the flow.

2. Pathlines — A pathline can be formed from fluid particles of different colour originated from the same points, such as a line formed after the introduction of ink into a shallow water flow.

3. Streaklines — A streakline represents a locus made by a miniature particles or tracers that passes at a same point.

1.4 Flow Patterns

1.4 Flow Patterns

Pathlines

1.4 Flow Patterns

1.5 Density• Density of a fluid, , • Definition: mass per unit volume, slightly affected by changes in temperature and

pressure. •

• = mass/volume = m/• • Units: kg/m3•

• Typical values:• Water = 1000 kg/m3; Air = 1.23 kg/m3

1.6 Viscosity• Viscosity, , is a measure of resistance to fluid flow as a

result of intermolecular cohesion. In other words, viscosity can be seen as internal friction to fluid motion which can then lead to energy loss.

• Different fluids deform at different rates under the same shear stress. The ease with which a fluid pours is an indication of its viscosity. Fluid with a high viscosity such as syrup deforms more slowly than fluid with a low viscosity such as water. The viscosity is also known as dynamic viscosity.

Units: N.s/m2 or kg/m/s Typical values:

Water = 1.14x10-3 kg/m/s; Air = 1.78x10-5 kg/m/s

Example:AirWaterOilGasolineAlcoholKeroseneBenzeneGlycerine

Fluid Newton’s law

of viscosity

Newtonian fluids obey refer

Newton’s’ law of viscosity is given by;

dy

du (1.1)

•The viscosity is a function only of the condition of the fluid, particularly its temperature.

•The magnitude of the velocity gradient (du/dy) has no effect on the magnitude of .

= shear stress = viscosity of fluiddu/dy = shear rate, rate of strain or velocity gradient

Newtonian and Non-Newtonian Fluid

Fluid Newton’s law

of viscosity

Non- Newtonianfluids

Do not obey

• The viscosity of the non-Newtonian fluid is dependent on the velocity gradient as well as the condition of the fluid.

Newtonian Fluids a linear relationship between shear stress and the velocity gradient

(rate of shear), the slope is constant the viscosity is constant

non-Newtonian fluids slope of the curves for non-Newtonian fluids varies

Newtonian and Non-Newtonian Fluid

If the gradient m is constant, the fluid is termed as Newtonian fluid. Otherwise, it is known as non-Newtonian fluid. Fig. 1.5 shows several Newtonian and non-Newtonian fluids.

Newtonian and Non-Newtonian Fluid

Definition: is the ratio of the viscosity to the density;

• will be found to be important in cases in which significant viscous and gravitational forces exist.

Units: m2/s

Typical values: Water = 1.14x10-6 m2/s; Air = 1.46x10-5 m2/s;

In general, viscosity of liquids with temperature, whereas

viscosity of gases with in temperature.

/

Kinematic viscosity,

Specific WeightSpecific weight of a fluid, • Definition: weight of the fluid per unit volume • Arising from the existence of a gravitational force • The relationship and g can be found using the

following:

Since = m/therefore = g (1.3)

Units: N/m3

Typical values:Water = 9814 N/m3; Air = 12.07 N/m3

The specific gravity (or relative density) can be defined as:

Definition 1: A ratio of the density of a liquid to the density of water at (4C, 1 atm), or

Unit: dimensionless.

Specific Gravity

STPwater

liquid

STPwater

liquidSG@@

STPair

gas

STPair

gasSG@@

Example 1.2•A reservoir of oil has a mass of 825 kg. The reservoir has a volume of 0.917 m3. Compute the density, specific weight, and specific gravity of the oil.

•Solution:

•

•3/900

917.0

825mkg

m

volume

massoil

3oil m/N882981.9x900g

mg

volume

weight

9.01000

900

@

STPw

oiloilSG

1.7 Surface Tension• Surface tension coefficient s can be defined as the

intensity of intermolecular traction per unit length along the free surface of a fluid, and its SI unit is N/m.

• The surface tension effect is caused by unbalanced cohesion forces at fluid surfaces which produce a downward resultant force which can physically seen as a membrane.

• The coefficient is inversely proportional to temperature and is also dependent on the type of the solid interface.

• For example, a drop of water on a glass surface will have a different coefficient from the similar amount of water on a wood surface.

1.7 Surface Tension• The effect may be becoming significant for small fluid

system such as liquid level in a capillary, as depicted in Fig. 1.6, where it will decide whether the interaction form by the fluid and the solid surface is wetted or non-wetted.

• If the adhesion of fluid molecules to the adjacent solid surface is stronger than the intermolecular cohesion, the fluid is said to wet on the surface. Otherwise, it is a non-wetted interaction.

1.7 Surface Tension The pressure inside a drop of fluid can be calculated using a free-

body diagram of a spherical shape of radius R cut in half, as shown in Fig. 1.7, and the force developed around the edge of the cut sphere is 2R.

This force must be balance with the difference between the internal pressure pi and the external pressure pe acting on the circular area of the cut. Thus,

2R = pR2

p = pi –pe =2

R

1.8 Vapour Pressure• Vapour pressure is the partial pressure produced by fluid

vapour in an open or a closed container, which reaches its saturated condition or the transfer of fluid molecules is at equilibrium along its free surface.

• In a closed container, the vapour pressure is solely dependent on temperature. In a saturated condition, any further reduction in temperature or atmospheric pressure below its dew point will lead to the formation of water droplets.

• On the other hand, boiling occurs when the absolute fluid pressure is reduced until it is lower than the vapour pressure of the fluid at that temperature.

• For a network of pipes, the pressure at a point can be lower than the vapour pressure, for example, at the suction section of a pump. Otherwise, vapour bubbles will start to form and this phenomenon is termed as cavitation.

Summary• This chapter has summarized on the aspect below: Understanding concept of a fluid Ways to visualise the flows and the continuity

nature of the fluid flow Fluid properties of density, specific weight, specific

gravity and viscosity were discussed. a Newtonian fluid against a non-Newtonian fluid, an

in-viscid flow against a viscous flow, and a wetted surface against a non-wetted surface.

Discussion on the surface tension and vapour pressure